A. Mashtakov, R. Duits, Yu. Sachkov, E. J. Bekkers, I. Beschastnyi. Tracking of Lines in Spherical Images via Sub-Riemannian Geodesics in SO(3) // Journal of Mathematical Imaging and Vision, 58(2), pp. 239–264, 2017.
In order to detect salient lines in spherical images, we consider the problem of minimizing the functional ∫_0^l C(γ(s))√(ξ^2+k_g^2(s))ds for a curve γ on a sphere with fixed boundary points and directions. The total length l is free, s denotes the spherical arclength, and kg denotes the geodesic curvature of γ. Here the smooth external cost C≥δ>0 is obtained from spherical data. We lift this problem to the sub-Riemannian (SR) problem in Lie group SO(3) and show that the spherical projection of certain SR geodesics provides a solution to our curve optimization problem. In fact, this holds only for the geodesics whose spherical projection does not exhibit a cusp. The problem is a spherical extension of a well-known contour perception model, where we extend the model by Boscain and Rossi to the general case ξ>0, C≠1. For C=1, we derive SR geodesics and evaluate the first cusp time. We show that these curves have a simpler expression when they are parameterized by spherical arclength rather than by sub-Riemannian arclength. For case C≠1 (data-driven SR geodesics), we solve via a SR Fast Marching method. Finally, we show an experiment of vessel tracking in a spherical image of the retina and study the effect of including the spherical geometry in analysis of vessels curvature.